A feedback control system is commonly employed to control the operation of a plant to achieve and maintain its desired output characteristic in response to the applied command or reference signal. A conventional feedback control system, FIG. 1, measures a parameter which is representative of the output of the system and exploits changes in the detected parameter to vary operational characteristics of the plant or system to regulate the system and provide a desired output. The feedback control system includes a compensator, connected within a continuous feedback loop for modifying the signal to appropriately vary the operational parameters of the system via an actuator. The major goals for the feedback control system are to minimize the effect of disturbances at the output of the system, and to minimize sensitivity of the closed loop response to plant parameter variations. To satisfy these requirements, the feedback of the system, properly weighted in frequency, must be maximized. These constraints uniquely define the optimal transfer function for the feedback loop. The purpose for the compensator of the feedback system is to implement a loop response reasonably close to the optimal. Often, a prefilter is employed prior to the feedback loop to appropriately filter the command signals to enable the system to respond more precisely to the command.
A commonly-used compensator employed in feedback control systems is a proportional-integral-derivative (PID) compensator which provides for varying degrees of gain and phase shift of the signal according to the frequency of the signal. In the prior art feedback system of FIG. 1, the PID compensator is characterized by a transfer function I/s+P+Ds, where s is a Laplace transform variable. R(s) represents the transfer function of a prefilter. The scalar parameters P, I, and D and the prefilter transfer function R(s) are tuned for optimal performance.
FIG. 2 provides Bode diagrams for transfer functions of each path of the compensator and for the entire compensator. The Bode diagrams provide a frequency domain representation of the gain in decibels as a function of frequency. FIG. 2 illustrates that proportional component P dominates at midrange frequencies, integral component I/s at lower frequencies, and derivative component Ds at higher frequencies.
The transfer functions which mathematically characterize the effect of the PID compensator are represented, in accordance with conventional mathematical notation, by the location of various transfer function poles and zeroes in the complex Laplace plane. The conventional PID compensator transfer function typically has two real zeros, as shown in FIG. 3. At the frequencies of the zeros, which correspond to the crossing points of the two asymptotes in FIG. 2, the total feedback gain provided by the compensator is about 3 dB higher than the gain of each of the two components. Typically, the P term dominates near f.sub.b, the Ds term dominates at frequencies over 4f.sub.b, and the I/s term dominates at frequencies up to f.sub.b /4, where f.sub.b is the crossover frequency at which loop gain is 0 dB.
After the feedback is maximized and the closed loop response from a summer of the system to the output of the system is calculated, the prefilter transfer function R(s) is chosen to yield the desired output response to a certain command.
A theoretically optimal loop response has been determined by Bode. For the purpose of industrial control, a simplified suboptimal Bode loop response can be employed. The suboptimal response is illustrated in FIG. 4 by a solid line. The slope of this suboptimal gain response is about -10 dB/octave. The transcendental loop transfer function which characterizes the suboptimal response can be closely approximated by a rational function.
As can be seen from FIG. 4, rather sharp corners occur at the sides of the Bode step. Any smoothing of the corners, especially the left one, caused by an improper or inaccurate rational function approximation, reduces the available feedback, resulting in reduced performance.
A typical loop gain Bode diagram of the system with a PID compensator is shown in FIG. 4. When provided with the same stability margin and the same average loop gain as an optimal Bode controller, the crossover frequency f.sub.b of the PID controller is about one-half that of the optimal Bode loop response. The feedback at frequency f.sub.b /4 is about 10 dB lower than that of a simplified Bode controller.
The conventional PID controller illustrated in FIG. 1 is in common use. When applied to a great variety of plants, the PID controller is easy to tune to provide robust and fairly good performance. However, the performance is not optimal.